At about 10:30P.M p.m., after around one hour's shadowing English I felt a little sleepy, but I didn't go to bed aso soon / then because I was afraid of waking up in the middle night,s. So I began to scroll Instagram and found it was really interesting.

This is not a grammar correction.
I noticed this was a really long sentence, so I broke it into two sentences where there was a change in topic.

About half a hour later, I became awared it was time to go to bed.

"aware" is not a verb. The verb you would use here is "realized"

But atby that time, I was not sleepy at all and couldn't help my mind woandering until 2:00 AM.

The set phrase "by that time" works well here. It is used after something has been going on for a while.
The verb in the idiom is "wander" You could also say "I couldn't keep my mind from wandering."

I had to picked up my smartphone and begain to listen to political stories to stop my mind.

I think this is what you mean. The correction I made shows that you intentionally did this as a solution to the problem--after staying up later didn't work.

Slowly I felt moere and more sleepy and felt inl a sleep.

I suffered from insomnia again last night, and it was really painful.

At about 10:30P.M p.m. after around one hour's of shadowing English, I felt a little sleepy, but I didn't go to bed as soonimmediately because I was afraid of waking up in the middle of the night, so I began to scroll Instagram and found it was really interesting.

About half an hour later, I awarrealized it was time to go to bed.

But at that time I was not sleepy at all and couldn't help my mind wofrom wandering until 2:00 AMa.m.

I had to picked up my smartphone and I began to listen to political stories to stop my mind.

Slowly, I felt moere and more sleepy and felt inl a sleep.

At about 10:30P.M PM, after around one hour's shadowing English, I felt a little sleepy, but I didn't want to go to bed aso soon because I was afraid of waking up in the middle night, so I began to scroll Instagram and found it was really interesting.

About half an hour later, I awaredbecame aware that it was time to go to bed.

"I realized" sounds more natural than "I became aware"

But at that time I was not sleepy at all and couldn't helkeep my mind from wondering until 2:00 AM.

I had to picked up my smartphone and began to listen to political stories to stop my mind.

Slowly I felt moere and more sleepy and felt inl a sleep.

At about 10:30P.M, after around one hour's of shadowing English, I felt a little sleepy, but I didn't go to bed as soon because I was afraid of waking up in the middle of the night, so I began to scroll Instagram and found it was really interesting.

About half a hour later, I awared it was time to go to bed.

you became aware (present tense)

I had to picked up my smartphone and began to listen to political stories to stop my mind.

Slowly I felt moere and more sleepy and felt inl a sleep.